Weak Lensing 3

Tom Kitching

Introduction• Scope of the lecture

• Power Spectra of weak lensing

• Statistics

Recap• Lensing useful for • Dark energy• Dark Matter

• Lots of surveys covering 100’s or 1000’s of square degrees coming online now

Recap• Lensing equation

• Local mapping • General Relativity relates this to the gravitational

potential

• Distortion matrix implies that distortion is elliptical : shear and convergence

• Simple formalise that relates the shear and convergence (observable) to the underlying gravitational potential

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Recap • Observed galaxies have instrinsic ellipticity and

shear • Reviewed shape measurement methods • Moments - KSB • Model fitting - lensfit

• Still an unsolved problem for largest most ambitous surveys

• Simulations • STEP 1, 2 • GREAT08 • Currently LIVE(!) GREAT10

Part V : Cosmic Shear

• Introduction to why we use 2-point stats

• Spherical Harmonics

• Derivation of the cosmic shear power spectra

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• When averaged over sufficient area the shear field has a mean of zero

• Use 2 point correlation function or power spectra which contains cosmological information

• Correlation function measures the tendency for galaxies at a chosen separation to have pre- ferred shape alignment

Spherical Harmonics• We want the 3D power spectrum for cosmic

shear • So need to generalise to spherical harmonics for

spin-2 field

• Normal Fourier Transform

• Want equivalent of the CMB power spectrum

• CMB is a 2D field• Shear is a 3D field

Spherical Harmonics

Describes general transforms on a sphere for any spin-weight quantity

Spherical Harmonics• For flat sky approximation and a scalar field

• Covariances of the flat sky coefficients related to the power spectrum

Derivation of CS power spectrum• The shear field we can observe is a 3D spin-2

• Can write done its spherical harmonic coefficients • From data :

• From theory :

Derivation of CS power spectrum• How to we theoretically predict ( r )?

• From lecture 2 we know that shear is related to the 2nd derivative of the lensing potential

• And that lensing potential is the projected Netwons potential

Derivation of CS power spectrum• Can related the Newtons potential to the

matter overdensity via Poisson’s Equation

Derivation of CS power spectrum• Generate theoretical shear estimate:

• Simplifies to

• Directly relates underlying matter to the observable coefficients

Derivation of CS power spectrum• Now we need to take the covariance of this to

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Large Scale Structure

Geometry

Tomography• What is “Cosmic Shear Tomography” and how

does it relate to the full 3D shear field?

• The Limber Approximation • (kx,ky,kz) projected to (kx,ky)

Tomography• Limber ok at small scales

• Very useful Limber Approximation formula (LoVerde & Afshordi)

Tomography

• Limber Approximation (lossy)

• Transform to Real space (benign)

• Discretisation in redshift space (lossy)

• Tomography • Generate 2D shear correlation in redshift bins• Can “auto” correlate in a bin• Or “cross” correlate between bin pairs• i and j refer to redshift bin pairs

Part VI : Prediction

• Fisher Matrices

• Matrix Manipulation

• Likelihood Searching

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What do we want?• How accurately can we estimate a model

parameter from a given data set?

• Given a set of N data point x1,…,xN

• Want the estimator to be unbiased • Give small error bars as possible

• The Best Unbiased Estimator

• A key Quantity in this is the Fisher (Information) Matrix

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What is the (Fisher) Matrix?• Lets expand a likelihood surface about the

maximum likelihood point

• Can write this as a Gaussian

• Where the Hessian (covariance) is

What is the Fisher Matrix?• The Hessian Matrix has some nice properties

• Conditional Error on

• Marginal error on

What is the Fisher Matrix?• The Fisher Matrix defined as the expectation of

the Hessian matrix

• This allows us to make predictions about the performance of an experiment !

• The expected marginal error on

Cramer-Rao • Why do Fisher matrices work?

• The Cramer-Rao Inequality : • For any unbiased estimator

The Gaussian Case• How do we calculate Fisher Matrices in

practice?

• Assume that the likelihood is Gaussian

The Gaussian Case

derivativematrix identity

derivative

How to Calculate a Fisher Matrix

• We know the (expected) covariance and mean from theory

• Worked example y=mx+c

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Adding Extra Parameters• To add parameters to a Fisher Matrix

• Simply extend the matrix

Combining Experiments• If two experiments are independent then the

combined error is simply

Fcomb=F1+F2

• Same for n experiments

Fisher Future Forecasting• We now have a tool with which we can predict

the accuracy of future experiments!

• Can easily • Calculate expected parameter errors• Combine experiments• Change variables• Add extra parameters

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• For shear the mean shear is zero, the information is in the covariance so (Hu, 1999)

• This is what is used to make predictions for cosmic shear and dark energy experiments

• Simple code available http://www.icosmo.org

Weak Lensing Surveys• Current and on going surveys

05 10 15 20

CFHTLenS**

Pan-STARRS 1**

Euclid

** complete or surveying * first light

Dark Energy• Expect constraints of 1% from Euclid

things we didn’t cover• Systematics • Photometric redshifts • Intrinsic Alignments

• Galaxy-galaxy lensing • Can use to determine galaxy-scale properties and

cosmology

• Cluster lensing• Strong lensing• Dark Matter mapping • ….• ….

Conclusion• Lensing is a simple cosmological probe • Directly related to General Relativity • Simple linear image distortions

• Measurement from data is challenging • Need lots of galaxies and very sophisticated

experiments

• Lensing is a powerful probe of dark energy and dark matter

Weak Lensing 3

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